An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.

Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ+.

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_2, \mu)}$$\end{document} -ITP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_3, \mu')}$$\end{document} -ITP hold, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu\geq \aleph_2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu'\geq \aleph_3}$$\end{document}. (shrink)

The typical rules for truth-trees for first-order logic without functions can fail to generate finite branches for formulas that have finite models–the rule set fails to have the finite tree property. In 1984 Boolos showed that a new rule set proposed by Burgess does have this property. In this paper we address a similar problem with the typical rule set for first-order logic with identity and functions, proposing a new rule set that does have the finite tree property.

Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

We show that given ù many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω²+1 and the SCH fails at ℵ ω².

In this paper, we study the notion of arborescent ordered sets, a generalizationof the notion of tree-property for cardinals. This notion was already studied previously in the case of directed sets. Our main result gives a geometric condition for an order to be ℵ0-arborescent.

Assume \ is a singular limit of \ supercompact cardinals, where \ is a limit ordinal. We present two methods for arranging the tree property to hold at \ while making \ the successor of the limit of the first \ measurable cardinals. The first method is then used to get, from the same assumptions, the tree property at \ with the failure of SCH at \. This extends results of Neeman and Sinapova. The second method is also used to (...) get the tree property at the successor of an arbitrary singular cardinal, which extends some results of Magidor–Shelah, Neeman and Sinapova. (shrink)

We say that a regular cardinal κ,, has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal,, is consistent with an arbitrary continuum function below which satisfies,. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal,, is consistent with an (...) arbitrary continuum function below which satisfies,. Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ. (shrink)

We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $. The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$.

We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor—Shelah [7].

We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA (...) with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω 1 ) for all λ ≥ ω 2. (shrink)

We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...) indeed [Formula: see text]. (shrink)

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa $ is strongly compact if and only if the strong tree property holds at $\kappa $, and supercompact if and only if ITP holds at $\kappa $. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; (...) these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph _n$ for all $2 \leq n < \omega $ simultaneously with the strong tree property at $\aleph _{\omega +1}$ ; we also show that it is consistent for ITP to hold at $\aleph _n$ for all $3 < n < \omega $ and at $\aleph _{\omega +1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously. (shrink)

We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...) always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

In this paper, we study the notions related to tree property 1 , or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in (...) Džamonja and Shelah [5] is crucially used throughout the paper. (shrink)

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left} \right]$ the tree property at $$\lambda = \left^{V\left[ {\left} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left} \right]$, where ${\rm{Add}}\left$ is the Cohen forcing for adding λ-many subsets of κ and $\left$ is the standard Mitchell forcing for (...) obtaining the tree property at $\lambda = \left^{V\left[ {\left} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense. (shrink)

We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm ZF} + \neg{\rm AC}_\omega}$$\end{document} + Every successor cardinal is regular + Every limit cardinal (...) is singular + Every successor cardinal satisfies the tree property” follows from hypotheses stronger in consistency strength than a supercompact limit of supercompact cardinals. A lower bound in consistency strength is provided by a result of Busche and Schindler, who showed that the consistency of the theory “ZF + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” implies the consistency of ADL(R). (shrink)

Fontanella :193–207, 2014) showed that if \ is an increasing sequence of supercompacts and \, then the strong tree property holds at \. Building on a proof by Neeman, we show that the strong tree property at \ is consistent with \, where \ is singular strong limit of countable cofinality.

In this paper, we investigate a new model theoretical tree property (TP), called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and [Formula: see text]-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples (...) of ATP and NATP, including pure groups, pure fields and valued fields. More precisely, we prove Mekler’s construction for groups, Chatzidakis’ style criterion for pseudo-algebraically closed (PAC) fields, and the AKE-style principle for valued fields preserving NATP. We give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras. (shrink)

We prove various theorems about the preservation and destruction of the tree property at ω2. Working in a model of Mitchell [9] where the tree property holds at ω2, we prove that ω2 still has the tree property after ccc forcing of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\aleph_1}$$\end{document} or adding an arbitrary number of Cohen reals. We show that there is a relatively mild forcing in this same model which destroys the tree property. Finally we (...) prove from a supercompact cardinal that the tree property at ω2 can be indestructible under ω2-directed closed forcing. (shrink)

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if $\delta$ has the tree property then $\delta^{+K} = \delta^+$ and $\delta$ is weakly compact in K.

We show that it is consistent, relative to ω many supercompact cardinals, that the super tree property holds at for all but there are weak square and a very good scale at.

If ω n has the tree property for all $2 \leq n and $2^{ , then for all X ∈ H ℵ ω and $n exists.

If $\omega_n$ has the tree property for all $2 \leq n < \omega$ and $2^{<\aleph_{\omega}} = \aleph_{\omega}$, then for all $X \in H_{\aleph_{\omega}}$ and $n < \omega, M^#_n$ exists.

In this paper we prove the equiconsistency of "Every ω 1 -tree which is first order definable over (H ω 1 ·ε) has a cofinal branch" with the existence of a Π 1 1 reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.

We prove the NTP1 property of a geometric theory T is inherited by theories of lovely pairs and H‐structures associated to T. We also provide a class of examples of nonsimple geometric NTP1 theories.

We show that the consistency of the theories “ holds below ” + “there is an injection ” + “both and fail” and + “ holds below ” + “there is an injection ” + “ satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long‐standing open questions concerning, weak square, approachability, and the tree property. There is nothing special about, and the injection into can be from any ordinal λ (...) (which means that pcf theory can fail badly in the models constructed if ). (shrink)